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Course Notes, March 11, 2009

1. $E(Y(n)) = \mu_x\sum_m^{} h(m)$ ind of n

2. $C_{xy}(m,n) = h*r_{xx}(n-m), only depends on n-m$

3. $r_{yy}(m,n) = h\_*C_{xy}(m-n) = h * C_{yx}(n-m)$

because: $r_{yy}(m,n) = E(Y(m)Y(n))$
$= E(Y(m)\sum_k^{}h(n-k)X(k))$
$= \sum_k^{}h(n-k)E(Y(m)X(k))$
$= \sum_k^{}h(n-k)C_{yx}(k-m)$
let $l = k-m$
$= \sum_l^{}h(n-m-l)C_{yx}(l) = h*C_{yx}(n-m)$
$= h\_*C_{xy}(m-n)$

observe:

$r_{yy}(m,n)$ only depends on n-m
$\Rightarrow Y(n)$ is wss

Flow Diagram of autocorollation transformation

$r_{xx}(n) \Rightarrow h(n) \Rightarrow C_{xy}(n) \Rightarrow time reversal \Rightarrow C_{xy}(-n) = C_{yx}(n) \Rightarrow h(n) \Rightarrow r_{yy}(n)$

Example

$Y(n) = X(n) + X(n-1)$
X(n) is i.i.d Gausian 0 mean with variance $\sigma_x^2$
Lets check if its wss
1.$E(X(n)) = 0, \forall_n$
2.$r_{xx}(m,n) = E(X(m)X(n)) = E(X(m)^2)$ if m=n else $E(X(m))E(X(n))$
$= \sigma^2, m = n ,else = 0$
$= \sigma^2\delta(m-n)$
$\Rightarrow r_{xx}(m,n) = r_xx{n-m}$
Now lets compute E(Y(n))
$E(y(n)) = E(X(n) + X(n-1))$
$= E(X(n)) + E(X(n-1))$
$= 0 + 0 = 0$
Now lets compute $r_{yy}(m)$
$r_{yy}(m) = h\_*C_{xy}(m)$
$= h\_*(h * r_{xx}(m))$
$= \sigma^2h\_*(h * \sigma^2\delta(m))$
$= \sigma^2(\delta(-n) + \delta(-n-1))*(\delta(n) + \delta(n-1))$
$= \sigma^2(\delta(n-1) + 2\delta(n) + \delta(n+1))$
...(erased too quickly, didnt get all of it)...

3.1.6 Estimating Correlation functions

suppose X(m) is wss process to estimate.
$r_{xx}(m) = E (X(n),X(n+m))$
any n would do. each n gives a sample

--Drestes 14:18, 11 March 2009 (UTC)

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